Strictly speaking, when \(A_i\)  and \(B_j\)  are non-commutative, due to the uncertainty principle, there are no definite outcomes of the two measurements are performed at the same time. The conditional probability defined in (REF ) or (REF ) needs to be extended to a two-time formulation of conditional probability. (REF ) or (REF ) is then considered as a special case when \([A_i, B_j]= 0\)  and the measurement times coincide. There are extensive research literature on how to construct two-time conditional probability in quantum theory [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}. One noticeable approach is based on the Page-Wootters' timeless formulation [5]}, [6]}, [7]}, [8]}, [9]}. In this work, we will continue to be based on the generalized Gleason theorem for POVM [15]}, [16]} to derive the two-time conditional probability, and leave the discussion of the Page-Wootters mechanism in Section IV.
